Lecture 1 Version: 14/08/28. Frontiers of Condensed Matter San Sebastian, Aug , Dr. Leo DiCarlo l.dicarlo@tudelft.nl dicarlolab.tudelft.

Transcription

1 Introduction to quantum computing (with superconducting circuits) Lecture 1 Version: 14/08/28 Frontiers of Condensed Matter San Sebastian, Aug , 2014 Dr. Leo DiCarlo l.dicarlo@tudelft.nl dicarlolab.tudelft.nl

2 What is quantum computing? A computation is a physical process. It may be performed by a piece of electronics or on an abacus, or in your brain, but it is a process that takes place in nature and as such it is subject to the laws of physics. Quantum computers are machines that rely on characteristically quantum phenomena, such as quantum interference and quantum entanglement in order to perform computation. - Artur Ekert

3 What is not quantum computing? It is tempting to say that a quantum computer is one whose operation is governed by the laws of quantum mechanics. But since the laws of quantum mechanics govern the behavior of all physical phenomena, this temptation must be resisted. Your laptop operates under the laws of quantum mechanics, but it is not a quantum computer. - N. David Mermin

4 Quantum information processing in a sentence Processing and communication of information based on control and measurement of quantum bits. superposition entanglement Why? Speedups/efficiency in algorithms (searching, factoring, simulating quantum systems) Absolute security of communication (quantum mechanical information cannot be copied) Testbed of quantum mechanics (measurement problem, local hidden variables, etc.)

5 A Turing machine is a universal model for computation. Computer Science Universal bounds on channel capacity, compressibility. Information theory Quantum information processing Quantum mechanics Thermodynamics Quantum systems can be in superposition, even entangled. Computing need not cost energy, erasing does.

6 The 2012 Nobel Prie Serge Haroche David Wineland 2012 Physics Nobel Prie Both Laureates work in the field of quantum optics studying the fundamental interaction between light and matter, a field which has seen considerable progress since the mid-1980s. Their ground-breaking methods have enabled this field of research to take the very first steps towards building a new type of super fast computer based on quantum physics. Perhaps the quantum computer will change our everyday lives in this century in the same radical way as the classical computer did in the last century. Announcement 2012 Nobel Prie

7 Current industry interest worldwide

8 and a lot of it in the Netherlands! October 2 nd, 2013: Announcement of QuTech: The Dutch Quantum Technology Institute Based in TUD: TNW, EWI, TNO, NWO+, industrial partners

9 We already have quantum companies

10 Probably the most (in)famous one

11 Media attention is exploding!

12 How many useful quantum algorithms are there? ~50 algorithms with quantum speedup, but most people know 2.

13 Numerous approaches to realiing quantum bits Nuclear Spins (NMR) Photons NV centers Trapped Ions 2012 Superconducting circuits

14 A few (provocative) statements Everybody really knows if you are ever going to make a real quantum computer, it must be solid state. -Benjamin Schumacher coined the term qubit Specifying the quantum state of a quantum computer with 200 qubits would require more classical bits than there are atoms in the universe. Factoring a technologically-relevant number (2000 bits) with a quantum computer at error rates 1/10 th the threshold and present clock speed will require 1 billion qubits operating for about 1 day. Today s most advanced quantum computer has 14 qubits. Today s most advanced solid-state quantum computer has 5 qubits. A useful quantum computer will need a supercomputer at its side.

15 Development roadmap for quantum computing We are taking this step! So, where are we precisely? Review: Devoret & Schoelkopf, Science 339, 1169 (2013)

16 Outline of lectures Lecture 1: Preliminaries: 1 qubit, 2 qubits,.infinity! Lecture 2: Simple algorithms and error correction Lecture 3: QC with superconducting circuits Mermin Nielsen & Chuang Wiseman & Milburn

17 Outline of Lecture #1 θ Description of quantum states ϕ Quantum gates ψ in Uˆ ψ out Quantum measurement ψ ˆM m = λ j λ j Quantum games

18 In the beginning, there was the quantum bit A quantum bit, or qubit, is a quantum system in which the Boolean states 0 and 1 are represented by a prescribed pair of normalied and mutually orthogonal quantum states labeled as and, respectively. 0 1 These two states form a computational basis. Any other (pure) state of the qubit can be written as a superposition. ψ α 1 α 0 α 0 = α 1

19 1-Qubit states ψ α 1 α 0 α 0 = α 1 α0, α1 C α α1 = 1 ( i cos( θ / 2) 0 e sin( / 2) 1 ) i e δ ϕ θ ψ = + θ v Bloch ϕ Global phase consequences δ does not have measureable Bloch sphere

20 Visualiing 1-Qubit states 0

21 Visualiing 1-Qubit states 1

22 The Cardinal 1-Qubit states ( ) ( i ) ( + i ) ( + ) 1

23 Quantum registers A collection of n qubits is called a quantum register of sie n Typical assumption: information is stored in a register in binary form. For example, the number 3 is represented in a n=2 qubit register as 3 = 1 1 = 11 MSQ LSQ And 2 as 2 = 1 0 = 10 There are N=2 n states of this kind, representing all binary strings of length n or numbers from 0 to N -1.

24 2-Qubit states Ψ = α 11 + α 10 + α 01 + α α00, α01, α10, α11 C = 1 α α α α α α α α Global phase does not have measureable consequences Is there a simple geometric way to visualie two-qubit states in analogy to the Bloch vector? Unfortunately, no.

25 Two-qubit superpositions 1 1 ( ) = ( ) = ( 0 ) 1 1 ( ) = ( ) ( ) = ( + ) ( + )

26 When are two qubits entangled? Two qubits are entangled when their joint wavefunction cannot be separated into a product of individual qubit wavefunctions Ψ = ϕ ψ vs Ψ = ϕ ψ + ϕ ψ Some common terms: Unentangled = separable = product Entangled = non-separable = non-product state state

27 1-Qubit logic gates A quantum logic gate is a device which performs a fixed unitary operation on selected qubits in a fixed period of time. Ψin Uˆ Ψout Ψ = Uˆ Ψ out in Uˆ is a unitary operator: ˆˆ ˆ ˆ ˆ UU = U U = I Unitary orthogonal in, orthogonal out norm preserving

28 Left to right, and right to left! Ψin A B C Ψout time We draw circuits from left to right Ψ = CBA Ψ out in but we do math from right to left

29 Catalog of 1-Qubit gates (Part 1) Identity I Iˆ Pauli X X X ˆ Pauli Y Y Yˆ 0 i i 0 Pauli Z Z Z ˆ Hadamard H Hˆ Rotations R ( ) n θ ˆ ˆ Rn ˆ ( θ) = cos( θ / 2) I isin( θ / 2) n σ σ = XYZ ˆ, ˆ, ˆ { }

30 Visualiing the Hadamard gate Ψ H Ψ All 1-Qubit unitary operations correspond to a rotation on the Bloch sphere (to within a numerical phase factor).

31 2-Qubit gates Controlled-Not gates b b a a b C-NOT gr b a b a a C-NOT rg

32 2-Qubit gates without classical analogs Controlled-Phase b b ϕ iϕ a ( e ) ab a C PHASE ϕ iϕ e π = Z = Z

33 Building up a quantum circuit A quantum circuit is a combination of quantum logic gates whose computational steps are synchronied in time H H = I H X H = Z = H H The process of simplifying a quantum circuit and/or finding a quantum circuit that implements a particular unitary using a specific set of gates is called quantum compiling. = H H

34 The quantum Fourier transform A quantum circuit is a combination of quantum logic gates whose computational steps are synchronied in time Ψin H π 2 H QFT π π 4 2 π 8 π 4 H π 2 H Ψout N 1 N 1 i2πlk 1 N Ψ out = e l k Ψ N l= 0 k= 0 N 1 1 α l = e N k = 0 i2πlk N α k in

35 The quantum Fourier transform A quantum circuit is a combination of quantum logic gates whose computational steps are synchronied in time Ψin H π 2 π 4 π 8 H π 2 π 4 H π 2 H Ψout A quantum circuit is said to be fast, or efficient, if the number of elementary operations taken to execute it increases no faster than a polynomial function of the sie (in qubits) of the input. QFT requires n=n(n+1)/2 gates, so it is O(n 2 ). Constructing the QFT from another universal set of gates only affects the circuit sie by a multiplicative constant which does not affect the quadratic scaling.

36 Evaluation of boolean functions Consider a mapping (not necessarily reversible) from n to m bits x y = f( x) X x U f f( x) x y X U f x f( x)

37 Evaluation of boolean functions Consider a mapping (not necessarily reversible) from n to m bits x y = f( x) x y U f x ( y+ f x ) ( ) mod 2 m

38 Quantum arithmetic x 1 x 0 y Quantum sum x 1 x 0 ( y+ x + x ) 1 0 mod 2 x 1 x 0 y Quantum carry x 1 x 0 ( y+ xx ) 1 0 mod 2

39 An example: encoding a boolean function in a quantum unitary 2 ( ) mod 8, f x = x= xx 1 0 x with a 2-bit number x y U f x ( y+ f x ) ( ) mod 8 x 1 x 1 x 0 x 0 + xx 1 0 x1 1 0 x 0 xx 0 xx x x 0 xx x0

40 Example: encoding a boolean function in a quantumunitary f x 2 ( ) x mod 8, = with x {0,1} 2 x y U f x ( y+ f x ) ( ) mod 8 xx x0 + y 2 y 1 y 0 x y y xx y xy y x y 0 0 So we need a circuit that implements the unitary transformation: x 1 x 1 x 0 y 2 y 1 y 0 x 0 x y y xx y xy x y 0 0 y

41 Example: encoding a boolean function in a quantum unitary x 1 x 1 x 0 y 2 y 1 y 0 x 0 x y y xx y xy x y 0 0 y x 1 x 0 y 2 y 1 y 0

42 3-Qubit Gates: the Toffoli gate x x y y xy TOFFOLI Other names Other symbols Controlled-Controlled-X (C-C-X) Controlled-Controlled-Not (C-C-NOT) X

43 3-Qubit gates: conditional-conditional phase x y π x y ( 1) xy C-C-PHASE π iπ e Other names Other symbols Controlled-Controlled-Z (C-C-Z) Toffoli-Sign (TS) Z

44 Toffoli decomposition into 1- and 2-Qubit gates /4 R + π = /4 R π /4 R π /2 R + π H R π /4 /4 R + π /4 R π /4 R + π H

45 CC-Phase decomposition into 1- and 2-Qubit gates /4 R + π π = R π /4 /4 R + π /4 R π /4 R π /4 R + π /4 R π /2 R + π

46 0 /4 R + π i e π /8 0 0 R π /4 /4 R π /2 R + π 0 R π /4 /4 R + π /4 R π /4 R + π 0 /4 R + π i e π /8 0 1 R π /4 /4 R π /2 R + π 1 X /4 R π /4 R + π /4 X R π /4 R + π 1 /4 R + π i e + π /8 1 0 R π /4 /4 X R + π X /2 R + π i e π /4 0 X /4 R + π /4 R π X

47 1 /4 R + π i e + π /8 1 1 R π /4 /4 X R + π X /2 R + π i e + π /4 1 X /4 R + π /4 R + π /4 X X + R π X /4 R + π i /2 ( 1) e π

48 C-C-Phase with 6 2-Qubit CNOT gates U e iπ / It works!

49 Is this the most efficient decomposition? Problem 4.4: (Minimal Toffoli construction) (Research) (1) What is the smallest number of two qubit gates that can be used to implement the Toffoli gate? (2) What is the smallest number of one qubit gates and CNOT gates that can be used to implement the Toffoli gate? (3) What is the smallest number of one qubit gates and controlled-z gates that can be used to implement the Toffoli gate? Lanyon et al., Nature Phys (2008)

50 Quantum parallelism x 0 U f x f( x) Prepare a superposition of all possible x inputs: 0 0 N H N 1 U f N x= 0 1 x f( x) N 1 N x 0 x f( x) N N x= 0 x= 0

51 Putting f(x) in the quantum phase x 1 QFT U f QFT e i2 π f ( x)/ M 1 x Quantum phase kick-back Prepare a superposition of all possible x inputs: x 1 H N QFT U f QFT 1 1 N i2 π f ( x)/ M 1 N x= 0 e x

52 Enter the spoiler: quantum measurement ψ ˆM m λ j λ j = 2 with probability ψλ j where Mˆ λ j = λ λ j j This is called the Born Rule Average value of measurement: m = j λ j ψλ = ψ λj λj λj ψ j = ψ Mˆ ψ j 2

53 Wavefunction collapse under measurement ψ Ẑ m = + 1 m = 1 0 or 1 m = prob=cos ( θ /2) Ẑ θ m = 1 2 prob=sin ( θ /2) ˆ 2 2 1cos ( / 2) 1sin ( / 2) cos( ) m= Z =+ θ θ = θ 1

54 A simple interpretation of the Bloch vector ψ Ẑ Zˆ = cos( θ ) v Bloch ψ ˆX ˆX Ẑ θ ϕ Yˆ y Xˆ = sin( θ)cos( ϕ) x ψ Yˆ vbloch = Xˆ Yˆ Zˆ {,, } Yˆ = sin( θ)sin( ϕ)

55 How measurements of X & Y are typically done in the lab In the lab, often measurements can only be conveniently done for one measurement operator (i.e., one measurement basis). Other operators can be measured by pre-rotating the qubit before measurement, and counter rotating afterward. Example: To measure Y R xˆ π 2 Ẑ m R x ˆ π 2 Example: To measure X R y ˆ π 2 Ẑ m R yˆ π 2

56 Quantum state tomography Quantum state tomography (QST): The procedure of experimentally determining an unknown quantum state (from multiple copies of it) 0 U? 0 U 0 U ˆX +1,-1,-1,+1,-1,..+1, +1 Yˆ +1,+1,-1,-1,+1,..-1, +1 Ẑ -1,+1,+1,+1,-1,..-1, -1 ˆ estimate X ˆ estimate Y ˆ estimate Z ~10,000 measurements of each operator to estimate expected value to ~1%

57 Projective measurement of 1 Qubit in a 2-Qubit register Ψ ˆM 1 m1 = λ j 2 with probability α λ j j where Mˆ 1 λ j = λ j λ j ϕ j This is called the generalied Born Rule Without loss of generality, can write 2-qubit state Ψ = α λ ϕ j j j j where ϕ j are normalied but not necessarily orthogonal

58 Quantum games

59 Deutsch s problem f f f f balanced unbalanced Classical Problem: You are handed a black box with one of the functions programmed in, but you re not told which one. Determine if the function is balanced or unbalanced. f i Quantum version: You are handed a quantum black box with one of the ftns encoded into the unitary U fi. Determine if the function is balanced or unbalanced. f i

60 Deutsch s problem: is your coin fair? 0 f f f f unbalanced balanced Classical Problem: You are handed a black box with one of the functions f i programmed in, but you re not told which one. Determine if the function is balanced or unbalanced. Quantum version: You are handed a quantum black box with one of the ftns encoded into the unitary U fi. Determine if the function is balanced or unbalanced. f i

61 Deutsch s problem: classical approach Classically, we must make two calls of the black box to determine if the function is balanced/unbalanced. Basically, you have to evaluate the function for both inputs, and see.

62 Deutsch s problem: quantum approach f f f f U f1 U f2 U f3 U f x y X

63 Deutsch s problem: quantum approach Execute this sequence calling the quantum black box once. 0 H x U f x H Ẑ m 1 H y y f( x) m = +1 m = 1 function is unbalanced function is balanced

64 Deutsch s problem: quantum approach Execute this sequence calling the quantum black box once. 0 H x U f x H Ẑ m 1 H y y f( x) 01 ( ) ( 0 1 ) 0 ( 0 1 ) + 1 ( 0 1 )

65 Deutsch s problem: quantum approach Execute this sequence calling the quantum black box once. 0 H x U f x H Ẑ m 1 H y y f( x) f ( 0 1 ) ) ( 0 1 ) ( 1) ( 1 f (0) (1) ( (0) (1) 0 + ( 1) f f 1 ) ( 0 1 )

66 Deutsch s problem: quantum approach Execute this sequence calling the quantum black box once. 0 H x U f x H Ẑ m 1 H y y f( x) 0 1 ( 0 1 ) ( 1 ) if f(0) = f(1) 0 if f(0) f(1)

67 Summary of lecture #1 One-qubit gates are rotations on the Bloch sphere. Quantum superposition carries over to the multi-qubit setting. Superposition states that are not separable are entangled. Entanglement is quite generic in a multi-qubit register. C-NOT + one-qubit rotations are universal: any unitary operation on any number of qubits can be compiled into a quantum circuit using C-NOTs and one-qubit rotations. Quantum superposition enables parallelism: it is possible to evaluate a function for all inputs at once. A quantum measurement on a qubit disturbs its state. This disturbance depends on the measurement result. The result itself is probabilistic. Borne s rule for quantum measurement specifies possible measurement results, their probabilities, and the post-measurement state of the qubit in terms of the eigenvalues and corresponding eigenstates of a hermitian operator. A single measurement on a qubit cannot reveal its initial state. Many measurements on identical copies of the state are required to make an estimate. The generalied Born rule tells us how measurement of a qubit affects the state of other qubits in a register.

68 Tomorrow we will use this hardware to perform simple algorithms